Nonlinear Partial Differential Equations with Applications

Tomáš Roubíček (auth.)9783764372934, 3-7643-7293-1

This book primarily concerns quasilinear and semilinear elliptic and parabolic partial differential equations, inequalities, and systems. The exposition leads the reader through the general theory based on abstract (pseudo-) monotone or accretive operators as fast as possible towards the analysis of concrete differential equations, which have specific applications in continuum (thermo-) mechanics of solids and fluids, electrically (semi-) conductive media, modelling of biological systems, or in mechanical engineering. Selected parts are mainly an introduction into the subject while some others form an advanced textbook.

The second edition simplifies and extends the exposition at particular spots and augments the applications especially towards thermally coupled systems, magnetism, and more. The intended audience is graduate and PhD students as well as researchers in the theory of partial differential equations or in mathematical modelling of distributed parameter systems.

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The monograph contains a wealth of material in both the abstract theory of steady-state or evolution equations of monotone and accretive type and concrete applications to nonlinear partial differential equations from mathematical modeling. The organization of the material is well done, and the presentation, although concise, is clear, elegant and rigorous. (…) this book is a notable addition to the existing literature. Also, it certainly will prove useful to engineers, physicists, biologists and other scientists interested in the analysis of (…) nonlinear differential models of the real world.

(Mathematical Reviews)

Table of contents :
Front Matter….Pages i-xx
Preliminary general material….Pages 1-27
Front Matter….Pages 29-29
Pseudomonotone or weakly continuous mappings….Pages 31-93
Accretive mappings….Pages 95-114
Potential problems: smooth case….Pages 115-131
Nonsmooth problems; variational inequalities….Pages 133-170
Systems of equations: particular examples….Pages 171-198
Front Matter….Pages 199-199
Special auxiliary tools….Pages 201-211
Evolution by pseudomonotone or weakly continuous mappings….Pages 213-301
Evolution governed by accretive mappings….Pages 303-333
Evolution governed by certain set-valued mappings….Pages 335-349
Doubly-nonlinear problems….Pages 351-391
Systems of equations: particular examples….Pages 393-448
Back Matter….Pages 449-476